In my opinion an administrator for each league, a go to guy for leaders to talk to about their fixtures, and a team of referees for each league. Meaning this team of 3-4 refs' priority would be the league they have been assigned, they would serve as backup for the other leagues.
This way, if I was organising a match, I would have 3-4 people to talk to try and agree on a day. Since they are dedicated to my league we will work together to get to an agreeable date and time. If no resolution is possible then i would go to my league's administrator who's job would be to find a ref from the other leagues, this administrator is better connected and better positioned to sort it out quickly.
I am happy to develop this idea with anybody who is interested to discuss further.
I would keep the structured fixture list. It is something we should be aiming to achieve. If both leaders agree on a different week however, then so be it, that's common sense.